Optimal. Leaf size=236 \[ \frac {e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left (-\frac {5}{2},\frac {m-5}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {3 e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left (-\frac {3}{2},\frac {m-5}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}-\frac {4 e^5 (e \sin (c+d x))^{m-5}}{a^3 d (5-m)}+\frac {7 e^3 (e \sin (c+d x))^{m-3}}{a^3 d (3-m)}-\frac {3 e (e \sin (c+d x))^{m-1}}{a^3 d (1-m)} \]
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Rubi [A] time = 0.64, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3872, 2875, 2873, 2564, 14, 2577, 270} \[ \frac {e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left (-\frac {5}{2},\frac {m-5}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {3 e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left (-\frac {3}{2},\frac {m-5}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}-\frac {4 e^5 (e \sin (c+d x))^{m-5}}{a^3 d (5-m)}+\frac {7 e^3 (e \sin (c+d x))^{m-3}}{a^3 d (3-m)}-\frac {3 e (e \sin (c+d x))^{m-1}}{a^3 d (1-m)} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2564
Rule 2577
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) (e \sin (c+d x))^m}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac {e^6 \int \cos ^3(c+d x) (-a+a \cos (c+d x))^3 (e \sin (c+d x))^{-6+m} \, dx}{a^6}\\ &=-\frac {e^6 \int \left (-a^3 \cos ^3(c+d x) (e \sin (c+d x))^{-6+m}+3 a^3 \cos ^4(c+d x) (e \sin (c+d x))^{-6+m}-3 a^3 \cos ^5(c+d x) (e \sin (c+d x))^{-6+m}+a^3 \cos ^6(c+d x) (e \sin (c+d x))^{-6+m}\right ) \, dx}{a^6}\\ &=\frac {e^6 \int \cos ^3(c+d x) (e \sin (c+d x))^{-6+m} \, dx}{a^3}-\frac {e^6 \int \cos ^6(c+d x) (e \sin (c+d x))^{-6+m} \, dx}{a^3}-\frac {\left (3 e^6\right ) \int \cos ^4(c+d x) (e \sin (c+d x))^{-6+m} \, dx}{a^3}+\frac {\left (3 e^6\right ) \int \cos ^5(c+d x) (e \sin (c+d x))^{-6+m} \, dx}{a^3}\\ &=\frac {e^5 \cos (c+d x) \, _2F_1\left (-\frac {5}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {3 e^5 \cos (c+d x) \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {e^5 \operatorname {Subst}\left (\int x^{-6+m} \left (1-\frac {x^2}{e^2}\right ) \, dx,x,e \sin (c+d x)\right )}{a^3 d}+\frac {\left (3 e^5\right ) \operatorname {Subst}\left (\int x^{-6+m} \left (1-\frac {x^2}{e^2}\right )^2 \, dx,x,e \sin (c+d x)\right )}{a^3 d}\\ &=\frac {e^5 \cos (c+d x) \, _2F_1\left (-\frac {5}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {3 e^5 \cos (c+d x) \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {e^5 \operatorname {Subst}\left (\int \left (x^{-6+m}-\frac {x^{-4+m}}{e^2}\right ) \, dx,x,e \sin (c+d x)\right )}{a^3 d}+\frac {\left (3 e^5\right ) \operatorname {Subst}\left (\int \left (x^{-6+m}-\frac {2 x^{-4+m}}{e^2}+\frac {x^{-2+m}}{e^4}\right ) \, dx,x,e \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {4 e^5 (e \sin (c+d x))^{-5+m}}{a^3 d (5-m)}+\frac {e^5 \cos (c+d x) \, _2F_1\left (-\frac {5}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {3 e^5 \cos (c+d x) \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) (e \sin (c+d x))^{-5+m}}{a^3 d (5-m) \sqrt {\cos ^2(c+d x)}}+\frac {7 e^3 (e \sin (c+d x))^{-3+m}}{a^3 d (3-m)}-\frac {3 e (e \sin (c+d x))^{-1+m}}{a^3 d (1-m)}\\ \end {align*}
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Mathematica [F] time = 10.74, size = 0, normalized size = 0.00 \[ \int \frac {(e \sin (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \sin \left (d x + c\right )\right )^{m}}{a^{3} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \left (d x + c\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x +c \right )\right )^{m}}{\left (a +a \sec \left (d x +c \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,{\left (e\,\sin \left (c+d\,x\right )\right )}^m}{a^3\,{\left (\cos \left (c+d\,x\right )+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (e \sin {\left (c + d x \right )}\right )^{m}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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